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Igor Kriz Aleš Pultr Introduction to Mathematical Analysis Igor Kriz Alesˇ Pultr Introduction to Mathematical Analysis IgorKriz AlesˇPultr DepartmentofMathematics DepartmentofAppliedMathematics(KAM) UniversityofMichigan FacultyofMathematicsandPhysics AnnArbor,MI CharlesUniversity USA Prague CzechRepublic ISBN978-3-0348-0635-0 ISBN978-3-0348-0636-7(eBook) DOI10.1007/978-3-0348-0636-7 SpringerBaselHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2013941992 ©SpringerBasel2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerBaselAGispartofSpringerScience+BusinessMedia(www.birkhauser-science.com) To Sophie ToJitka Preface Thisbookis a resultof a long-termprojectwhichoriginatedin courseswe taught to undergraduatestudents who specialize in mathematics. These students had ı-" calculusbefore,buttheredidnotseemtobeasuitablecomprehensivetextbookfor afollow-upcourseinanalysis. We wanted to write such a textbook based on our courses, but that was not the onlygoal. Teachingbrightstudentsis aboutintroducingthem to mathematics. Therefore,wewantedtowriteabookwhichthestudentsmaywanttokeepafterthe courseisover,andwhichcouldservethemasabridgetohighermathematics.Such abookwouldnecessarilyexceedthescopeoftheircourses. We start with standard material of second year analysis: multi-variable differential calculus, Lebesgue integration, ordinary differential equations and vector calculus.Whatmakesallthisgosmoothlyisthatweintroducesomebasicconcepts of pointset topologyfirst. Since ouraim is to be completelyrigorousand as self- contained as possible, we also include a Preliminaries chapter on the basic topic of one-variablecalculus, and two Appendiceson the necessary concepts of linear algebra.Thisprettymuchcomprisesthefirstpartofourbook. Withthefoundationscovered,itispossibletoventuremuchfurther.Thecommon themeofthesecondpartofourbookistheinterplaybetweenanalysisandgeometry. After a second installmentof pointset topology,we are quicklyable to introduce complexanalysis, and after some multi-linearalgebra,also manifolds,differential forms and the general Stokes Theorem. The methods of manifolds and complex analysiscombineinatreatmentofRiemannsurfaces.Basicmethodsofthecalculus ofvariationsareappliedtoatheoryofgeodesics,whichinturnleadstobasictensor calculusandRiemanniangeometry.Finally,infinite-dimensionalspaces,whichhave alreadymadeanappearanceinmultipleplacesthroughoutthetext,aretreatedmore systematicallyinachapteronthebasicconceptsoffunctionalanalysis,andanother onafewofitsapplications. The totalamountof materialin this bookcannotbe coveredin anysingle year course. An instructor of a course based on this book should probably aim for covering the first part, and take his or her picks in the second part. As already mentioned, we hope to motivate the student to hold on to their textbook, and use it for further study in years to come. They will eventually get to more advanced books in analysis and beyond, but here they can get, relatively quickly, their first glimpseofabigpicture. vii viii Preface Becauseofthis,theaimofourbookisnotlimitedtoundergraduatestudents.This text may equally well serve a graduate student or a mathematician at any career stage who would like a quick source or reference on basic topics of analysis. A scientist (for example in physics or chemistry) who may have always been using analysisin theirwork,can usethisbooktogobackandfillin the rigorousdetails and mathematicalfoundations.Finally, an instructorof analysis, even if notusing this book as a textbook, may want to use it as a reference for those pesky proofs whichusuallygetskippedinmostcourses:wedoquiteafewofthem. AnnArbor,USA IgorKriz Prague1,CzechRepublic AlesˇPultr Contents PartI ARigorousApproachtoAdvancedCalculus 1 Preliminaries................................................................ 3 1 Realandcomplexnumbers.............................................. 3 2 ConvergentandCauchysequences ..................................... 10 3 Continuousfunctions.................................................... 11 4 DerivativesandtheMeanValueTheorem.............................. 13 5 Uniformconvergence.................................................... 18 6 Series.Seriesoffunctions............................................... 19 7 Powerseries.............................................................. 23 8 AfewfactsabouttheRiemannintegral ................................ 26 9 Exercises ................................................................. 30 2 MetricandTopologicalSpacesI.......................................... 33 1 Basics..................................................................... 33 2 Subspacesandproducts ................................................. 37 3 Sometopologicalconcepts.............................................. 39 4 Firstremarksontopology............................................... 43 5 Connectedspaces........................................................ 47 6 Compactmetricspaces.................................................. 51 7 Completeness ............................................................ 54 8 Uniform convergenceof sequences of functions. Application:Tietze’sTheorems......................................... 57 9 Exercises ................................................................. 62 3 MultivariableDifferentialCalculus ...................................... 65 1 Realandvectorfunctionsofseveralvariables ......................... 65 2 Partialderivatives.Definingtheexistenceofatotaldifferential ...... 66 3 Compositionoffunctionsandthechainrule........................... 71 4 Partialderivativesofhigherorder.Interchangeability ................. 74 5 TheImplicitFunctionsTheoremI:Thecaseofasingleequation .... 77 6 TheImplicitFunctionsTheoremII:Thecaseofseveralequations ... 81 7 Aneasyapplication:regularmappingsandtheInverse FunctionTheorem ....................................................... 86 ix